RAMANUJAN CONTEST 2021
UNIVERSITY OF MAURITIUS
Faculty of Science
Department of Mathematics
Answer All Questions
1. Find the smallest positive integer which is a cube, and whose ordinary decimal
digits can be rearranged into another cube. (Integers are written with no leading
zeros.)
P
2. Let xk = ark be the k-th term of a geometric series. Suppose ∞
k=0 xk = 1 and
P
∞
3
k=0 xk = 2. Determine r.
3. A small projectile is placed at the base of a hill with slope 12 . At what angle α
should it be launched so that it lands as far up the hill as possible? Express your
answer in terms of tan α.
4. Find the smallest integer n with the property that, whenever 10n = ab is written
as a product of two positive integers, then at least one of {a, b} contains the digit
“0”.
5. Find the prime factorization of the smallest integer n which is divisible by k k for
k ∈ {1, 2, 3, ..., 10}.
6. Determine all real numbers s such that the infinite series
∞
X
√
√
( n + 1 − n)s ,
n=1
converges.
7. Let z1 , z2 , and z3 be three complex numbers in geometric progression. Suppose
that the average of these numbers is 10, while the average of their squares is 20i.
Determine the value of z2 , the middle term.
8. The positive integer 5n has 7 positive factors (including 1 and itself). How many
positive factors does 2n have?
Please turn the page.
Date: September 28, 2021.
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RAMANUJAN CONTEST 2021
9. For how many positive integers n < 1000 is it the case that the remainders when
n is divided by 4, 5 or 6 are the same?
10. I have 6 coins, of which 5 are normal and one has heads on both sides. I pick
one at random and toss it 6 times and it comes up heads every time. What is the
chance that, if I toss it again, it will come up heads?
11. Show that there is a positive real number c such that
Z π
xc sin x dx = 3.
0
12. Suppose f (x) and g(x) are both solutions of the second-order differential equation
y 00 + p(x)y 0 + q(x)y = 0,
and suppose that (f (x))2 + (g(x))2 = 1 for all x. Express (f 0 (x))2 + (g 0 (x))2 and p(x)
in terms of q(x) and/or q 0 (x).
13. Find integers m and n so that
q
√
√
√
90 + 2 2021 = m + n.
14. Find all positive real-valued differentiable functions f with the property that for
all x,
Z x
2
(f (x)) =
(f (t))2 + (f 0 (t))2 dt + 2021.
0
END OF QUESTION PAPER